Measures of dispersion - range, quartile deviation, mean deviation, standard deviation, Coefficient of variation

MEASURES OF DISPERSION (CONCEPTUAL NOTES)

1. Introduction

Measures of dispersion show how spread out or scattered the data are.

While the mean, median, and mode tell us about the central value, dispersion tells us how far the data are from the center.

Two datasets can have the same mean but different spreads, so dispersion helps us understand variability or consistency.

Common measures:

Range

Quartile Deviation (Q.D.)

Mean Deviation (M.D.)

Standard Deviation (S.D.)

Coefficient of Variation (C.V.)

2. Range

Definition: The range is the difference between the largest and smallest values in the data.

Purpose: Gives a quick idea of how spread out the data is.

Advantages: Very simple and easy to calculate.

Disadvantages:

Uses only two values, ignoring the rest of the data.

Very sensitive to extreme values or outliers.

3. Quartile Deviation (Q.D.)

Definition: Quartile deviation measures the spread of the middle 50% of the data.

It focuses on the central part of the dataset, ignoring extreme values.

Purpose: To understand typical variability without being affected by very high or very low values.

Advantages: Less affected by outliers, useful for skewed data.

Disadvantages: Ignores the lower 25% and upper 25% of the data.

4. Mean Deviation (M.D.)

Definition: Mean deviation is the average amount by which each value differs from the central value (like mean or median).

Purpose: Measures overall variability considering all observations.

Advantages: Uses all data points; gives a better sense of average deviation.

Disadvantages: Can be harder to calculate manually; does not give as precise results as standard deviation for advanced statistical analysis.

5. Standard Deviation (S.D.)

Definition: Standard deviation shows the average distance of all data points from the mean, giving a clear idea of how clustered or spread out the data are.

Purpose: The most widely used measure of dispersion in statistics.

Advantages:

Considers every data point.

Useful in probability and advanced statistical analysis.

Disadvantages:

Sensitive to extreme values.

Slightly complicated to calculate manually.

6. Coefficient of Variation (C.V.)

Definition: The coefficient of variation is a relative measure of dispersion expressed as a percentage of the mean.

Purpose: Allows comparison of variability between datasets with different units or averages.

Advantages:

Unit-free, so it’s easy to compare data sets.

Useful for analyzing relative consistency.

Disadvantages:

Not meaningful if the mean is zero.

Cannot be used for qualitative data.

8. Conclusion

Measures of dispersion help us understand variability and reliability of data.

Range and Q.D. are simpler but less precise.

Mean deviation and standard deviation are more accurate and consider all data points.

Coefficient of variation is useful when comparing different datasets.

MCQs – Measures of Dispersion

1. Range

1. Range is defined as:

A) Difference between mean and median

B) Difference between maximum and minimum values

C) Difference between first and third quartiles

D) Average deviation from mean

✅ Answer: B

2. Which of the following is a disadvantage of the range?

A) Easy to calculate

B) Only uses two extreme values

C) Useful for large datasets

D) Shows overall variability

✅ Answer: B

2. Quartile Deviation (Q.D.)

3. Quartile deviation measures:

A) Spread of all data points

B) Spread of the middle 50% of data

C) Difference between max and min

D) Relative variability

✅ Answer: B

4. One advantage of quartile deviation is:

A) Considers all data points

B) Less affected by outliers

C) Simple to calculate

D) Provides exact variability

✅ Answer: B

5. Quartile deviation is also called:

A) Semi-interquartile range

B) Standard deviation

C) Mean deviation

D) Coefficient of variation

✅ Answer: A

3. Mean Deviation (M.D.)

6. Mean deviation is the:

A) Average difference of data points from the central value

B) Maximum minus minimum value

C) Spread of middle 50%

D) Square root of variance

✅ Answer: A

7. Mean deviation uses:

A) Only extremes of data

B) Only middle values

C) All observations

D) Only mean value

✅ Answer: C

4. Standard Deviation (S.D.)

8. Standard deviation is:

A) The difference between max and min

B) A measure of average spread from the mean

C) Only based on median

D) Spread of middle 50%

✅ Answer: B

9. One disadvantage of standard deviation is:

A) Does not use all data points

B) Sensitive to extreme values

C) Cannot be used for continuous data

D) Not widely used

✅ Answer: B

10. Standard deviation is preferred over mean deviation because:

A) It is less sensitive to outliers

B) It is easier to calculate

C) It has better mathematical properties for analysis

D) It ignores extremes

✅ Answer: C

5. Coefficient of Variation (C.V.)

11. Coefficient of variation is used to:

A) Compare absolute spread

B) Compare relative variability of two datasets

C) Measure only maximum values

D) Measure middle 50%

✅ Answer: B

12. C.V. is expressed in:

A) Units of data

B) Absolute numbers

C) Percentage

D) None of the above

✅ Answer: C

13. C.V. is not meaningful when:

A) Mean is large

B) Mean is zero or negative

C) Standard deviation is large

D) Median is small

✅ Answer: B

6. General Questions

14. Which of the following uses all data points?

A) Range

B) Quartile Deviation

C) Mean Deviation and Standard Deviation

D) None of the above

✅ Answer: C

15. Which measure is least affected by extreme values?

A) Range

B) Quartile Deviation

C) Standard Deviation

D) Mean Deviation

✅ Answer: B